3.605 \(\int x^{12} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{b^5 x^{23} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 \left (a+b x^2\right )}+\frac{5 a b^4 x^{21} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x^{19} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}+\frac{a^5 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{a^4 b x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{10 a^3 b^2 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )} \]

[Out]

(a^5*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*(a + b*x^2)) + (a^4*b*x^15*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2)) + (10*a^3*b^2*x^17*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4])/(17*(a + b*x^2)) + (10*a^2*b^3*x^19*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^
4])/(19*(a + b*x^2)) + (5*a*b^4*x^21*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(21*(a + b
*x^2)) + (b^5*x^23*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(23*(a + b*x^2))

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Rubi [A]  time = 0.189802, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b^5 x^{23} \sqrt{a^2+2 a b x^2+b^2 x^4}}{23 \left (a+b x^2\right )}+\frac{5 a b^4 x^{21} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x^{19} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}+\frac{a^5 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{a^4 b x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{10 a^3 b^2 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^12*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(a^5*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*(a + b*x^2)) + (a^4*b*x^15*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])/(3*(a + b*x^2)) + (10*a^3*b^2*x^17*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4])/(17*(a + b*x^2)) + (10*a^2*b^3*x^19*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^
4])/(19*(a + b*x^2)) + (5*a*b^4*x^21*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(21*(a + b
*x^2)) + (b^5*x^23*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(23*(a + b*x^2))

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Rubi in Sympy [A]  time = 26.6059, size = 207, normalized size = 0.81 \[ \frac{256 a^{5} x^{13} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{2028117 \left (a + b x^{2}\right )} + \frac{128 a^{4} x^{13} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{156009} + \frac{160 a^{3} x^{13} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{52003} + \frac{80 a^{2} x^{13} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{9177} + \frac{10 a x^{13} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{483} + \frac{x^{13} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

256*a**5*x**13*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(2028117*(a + b*x**2)) + 128*
a**4*x**13*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/156009 + 160*a**3*x**13*(a + b*x*
*2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/52003 + 80*a**2*x**13*(a**2 + 2*a*b*x**2
+ b**2*x**4)**(3/2)/9177 + 10*a*x**13*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x*
*4)**(3/2)/483 + x**13*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/23

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Mathematica [A]  time = 0.0387963, size = 83, normalized size = 0.33 \[ \frac{x^{13} \sqrt{\left (a+b x^2\right )^2} \left (156009 a^5+676039 a^4 b x^2+1193010 a^3 b^2 x^4+1067430 a^2 b^3 x^6+482885 a b^4 x^8+88179 b^5 x^{10}\right )}{2028117 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^13*Sqrt[(a + b*x^2)^2]*(156009*a^5 + 676039*a^4*b*x^2 + 1193010*a^3*b^2*x^4 +
1067430*a^2*b^3*x^6 + 482885*a*b^4*x^8 + 88179*b^5*x^10))/(2028117*(a + b*x^2))

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Maple [A]  time = 0.011, size = 80, normalized size = 0.3 \[{\frac{{x}^{13} \left ( 88179\,{b}^{5}{x}^{10}+482885\,a{b}^{4}{x}^{8}+1067430\,{a}^{2}{b}^{3}{x}^{6}+1193010\,{a}^{3}{b}^{2}{x}^{4}+676039\,{a}^{4}b{x}^{2}+156009\,{a}^{5} \right ) }{2028117\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/2028117*x^13*(88179*b^5*x^10+482885*a*b^4*x^8+1067430*a^2*b^3*x^6+1193010*a^3*
b^2*x^4+676039*a^4*b*x^2+156009*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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Maxima [A]  time = 0.699748, size = 77, normalized size = 0.3 \[ \frac{1}{23} \, b^{5} x^{23} + \frac{5}{21} \, a b^{4} x^{21} + \frac{10}{19} \, a^{2} b^{3} x^{19} + \frac{10}{17} \, a^{3} b^{2} x^{17} + \frac{1}{3} \, a^{4} b x^{15} + \frac{1}{13} \, a^{5} x^{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^12,x, algorithm="maxima")

[Out]

1/23*b^5*x^23 + 5/21*a*b^4*x^21 + 10/19*a^2*b^3*x^19 + 10/17*a^3*b^2*x^17 + 1/3*
a^4*b*x^15 + 1/13*a^5*x^13

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Fricas [A]  time = 0.25787, size = 77, normalized size = 0.3 \[ \frac{1}{23} \, b^{5} x^{23} + \frac{5}{21} \, a b^{4} x^{21} + \frac{10}{19} \, a^{2} b^{3} x^{19} + \frac{10}{17} \, a^{3} b^{2} x^{17} + \frac{1}{3} \, a^{4} b x^{15} + \frac{1}{13} \, a^{5} x^{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^12,x, algorithm="fricas")

[Out]

1/23*b^5*x^23 + 5/21*a*b^4*x^21 + 10/19*a^2*b^3*x^19 + 10/17*a^3*b^2*x^17 + 1/3*
a^4*b*x^15 + 1/13*a^5*x^13

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{12} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**12*((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.272574, size = 142, normalized size = 0.56 \[ \frac{1}{23} \, b^{5} x^{23}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{21} \, a b^{4} x^{21}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{19} \, a^{2} b^{3} x^{19}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{17} \, a^{3} b^{2} x^{17}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{3} \, a^{4} b x^{15}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{13} \, a^{5} x^{13}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^12,x, algorithm="giac")

[Out]

1/23*b^5*x^23*sign(b*x^2 + a) + 5/21*a*b^4*x^21*sign(b*x^2 + a) + 10/19*a^2*b^3*
x^19*sign(b*x^2 + a) + 10/17*a^3*b^2*x^17*sign(b*x^2 + a) + 1/3*a^4*b*x^15*sign(
b*x^2 + a) + 1/13*a^5*x^13*sign(b*x^2 + a)